How can I hide the source code that overflows a listing's frame?

Here is a version using the viewport option that package adjustbox provides for its box commands. The code had to be rearranged a bit such that the frame around the listings is added by \adjustbox and the captions by \captionof. Otherwise you would get a missing right side of the frame.

Everything was wrapped into a new environment codebox which takes the box width as its first and the listing's caption as its second parameter. It's basically just a sketch of the idea; a better interface for more customization could be added by providing a key-value option list for the environment.

Full example code:

\documentclass[a4paper,export]{report}
\usepackage{xcolor}
\usepackage{lipsum}
\usepackage{listings}
\usepackage{caption}
\usepackage{adjustbox}

\lstset{
    %numbers=left,numberstyle=\tiny, stepnumber=1, numbersep=5pt,
    breaklines=true,
    breakatwhitespace=false,
    tabsize=2,
    basicstyle=\footnotesize,%\ttfamily,
%    frame=single,
%    rulecolor=\color{gray}
}

\newenvironment{codebox}[2]{%
    \begin{minipage}[t]{#1}%
    \vspace*{-0.5\baselineskip}
    \captionof{lstlisting}{#2}\par
    \vspace*{-0.7\baselineskip}
    \begin{adjustbox}{
        cfbox=gray,
        clip=true,
        viewport={\fboxrule+\fboxsep} {-\depth} {\linewidth-\fboxrule-\fboxsep} {\height}
    }%
    \hbadness=100
    \hfuzz=\maxdimen
}{%
    \end{adjustbox}%
    \end{minipage}%
}

\begin{document}
\lipsum[1]

\noindent
\begin{codebox}{0.48\textwidth}{Code 1}
    \begin{lstlisting}[breaklines=false]
<?xml version="1.0" encoding="UTF-8"?>
<s:Envelope xmlns:s="http://schemas.xmlsoap.org/soap/envelope/">
    <s:Body>
    <GetSpecialtiesResponse xmlns="http://tempuri.org/">
    <GetSpecialtiesResult xmlns:a="http://schemas.datacontract.org/2004/07/WsAgenda.DataContracts" xmlns:i="http://www.w3.org/2001/XMLSchema-instance">
    <a:Specialty>
        <a:code>cardiology</a:code>
        <a:name>Cardiologie</a:name>
    </a:Specialty>
    <a:Specialty>
        <a:code>neurology</a:code>
        <a:name>Neurologie</a:name>
    </a:Specialty>
    <a:Specialty>
        <a:code>urology</a:code>
        <a:name>Urologie</a:name>
    </a:Specialty>
    <a:Specialty>
        <a:code>physiotherapy</a:code>
        <a:name>Physiotherapie</a:name>
    </a:Specialty>
    </GetSpecialtiesResult>
    </GetSpecialtiesResponse>
    </s:Body>
</s:Envelope>
    \end{lstlisting}
\end{codebox}
\hfill
\begin{codebox}{.48\textwidth}{Code 2}
    \begin{lstlisting}[breaklines=true]
{"Specialties": [
    {
    "id": "cardiology",
    "description": "Cardiologie"
    },
    {
    "id": "neurology",
    "description": "Neurologie"
    },
    {
    "id": "urology",
    "description": "Urologie"
    },
    {
    "id": "physiotherapy",
    "description": "Physiotherapie"
    }
]}
    \end{lstlisting}
\end{codebox}
\medskip

\lipsum[2]
\end{document}

outputs